{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-prob-lln",
    "area": "mathematics",
    "topic": "probability",
    "origin": "original",
    "title": "LLN versus CLT",
    "skills": [
      "law-of-large-numbers",
      "central-limit-theorem"
    ],
    "priority": "core",
    "difficulty": "easy",
    "estimated_minutes": 5,
    "prerequisites": [
      "expectation",
      "variance"
    ],
    "prompt": "For i.i.d. variables with mean $\\mu$ and finite variance $\\sigma^2$, state what the law of large numbers says about the sample mean $\\overline{X}_n$ and what the central limit theorem says about its standardized error.\n",
    "answer": "The law of large numbers says\n$$\n\\overline{X}_n \\xrightarrow{p} \\mu.\n$$\nThe central limit theorem says\n$$\n\\frac{\\sqrt{n}(\\overline{X}_n-\\mu)}{\\sigma}\n\\xrightarrow{d} \\mathcal{N}(0,1).\n$$\n",
    "check": {
      "kind": "symbolic",
      "id": "original-prob-lln",
      "values": [
        1
      ]
    }
  }
}